Flows

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Flows of vector fields: existence and uniqueness.

A reference for this material is Section 9 and Appendix D of John M. Lee. Introduction to smooth manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013. xvi+708pp. ISBN: 978-1-4419-9981-8.

Existence of flows: Euclidean case Let $\Omega \subset \mathbb {R}^n$ open and $V \in C^{\infty }(\Omega ; \mathbb {R}^n)$. Then for all $p \in \Omega $ there are $W \subset \Omega $ open containing $p$, $\varepsilon > 0 $, and $\Phi : W \times (- \varepsilon , \varepsilon ) \to \Omega $ smooth such that

For all $x \in W$, one has
\[ \Phi (x,0) = x . \]
For all $x \in W$ and $t \in (- \varepsilon , \varepsilon )$, one has
\[ \frac {\partial }{\partial t} \Phi (x,t) = V\left ( \Phi (x,t) \right ) . \]

The map $\Phi $ is called a local flow of $V$. For fixed $t \in (- \varepsilon , \varepsilon ) $, we sometimes denote the map $x \mapsto \Phi (x,t)$ by $\Phi _t$. For fixed $x \in W$, the curves $t \mapsto \Phi (x,t)$ are called the flow lines of $V$.

Flow lines are unique: Euclidean case Let $\Omega \subset \mathbb {R}^n$ open and $V \in C^{\infty }(\Omega ; \mathbb {R}^n)$. If $\gamma _1 , \gamma _2 : (-\varepsilon , \varepsilon ) \to \Omega $ satisfy $\gamma _1 (0) = \gamma _2 (0) $ and

\[ \gamma _i ' (t) = V(\gamma _i (t)) \]

for all $ i \in \{ 1 , 2\}$ and all $t \in (- \varepsilon , \varepsilon ) $, then $\gamma _1 = \gamma _2$.

The flow satisfies

\[ \Phi _{s + t} = \Phi _s \circ \Phi _t \]

for all $s,t \in (-\varepsilon / 2 , \varepsilon / 2)$.

Existence of flows For each $V \in \mathfrak {X} (M)$ and $p \in M$, there are $W \subset M$ open containing $p$, $\varepsilon > 0$, and $\Phi : W \times (- \varepsilon , \varepsilon ) \to M$ smooth such that

For all $x \in W$, one has
\[ \Phi (x,0) = x . \]
For all $x \in W $ and $t \in (- \varepsilon , \varepsilon )$, one has
\[ \frac {\partial }{\partial t} \Phi (x,t) = V(\Phi (x,t)) . \]

Solution:

Choose $(U, \varphi )$ a chart around $p$ and let $\hat {V} = (V^1 , \ldots , V^n) : \varphi (U) \to \mathbb {R}^n $ be given by

\[ V ( x ) = \sum _{i = 1 }^n V^i (\varphi (x)) \partial _i \vert _x . \]

Since $\varphi _{\ast } \partial _i = e _i $, we have $\varphi _{\ast } V(x) = \hat {V} (\varphi (x) )$ for all $x \in U$.

By Theorem thm:e-and-u, there are $ W_0 \subset \varphi (U)$ open containing $\varphi (p)$, $\varepsilon > 0 $, and $\Psi : W_0 \times (- \varepsilon , \varepsilon ) \to \varphi (U)$ smooth such that

For all $x \in W_0$, one has
\[ \Psi (x,0) = x . \]
For all $x \in W_0$ and $t \in (- \varepsilon , \varepsilon )$, one has
\[ \frac {\partial }{\partial t} \Psi (x,t) = \hat {V}\left ( \Psi (x,t) \right ) . \]

With this, define $W : = \varphi ^{-1} (W_0)$ and $\Phi : W \times (- \varepsilon , \varepsilon ) \to M$ as

\[ \Phi (x,t) : = \varphi ^{-1} ( \Psi ( \varphi (x) , t ) ). \]

For all $x \in W$, one has

\begin{align*} \Phi (x,0 ) & = \varphi ^{-1} ( \Psi ( \varphi (x) , 0 ) ) \\ & = \varphi ^{-1} ( \varphi (x) ) \\ & = x . \end{align*}

Also, for all $x \in W$ and $t \in (- \varepsilon , \varepsilon ) $, we have

\begin{align*} \frac {\partial }{\partial t} \Phi (x,t) & = \varphi ^{-1} _{\ast } ( \frac {\partial }{\partial t} \Psi ( \varphi (x) , t ) ) \\ & = \varphi ^{-1} _{\ast } ( \hat {V} ( \Psi (\varphi (x) , t ) ) \\ & = V ( \varphi ^{-1} ( \Psi ( \varphi (x) , t ) ) ) \\ & = V ( \Phi (x, t) ) . \end{align*}

Flow lines are unique Let $V \in \mathfrak {X} (M) $ and $\gamma _1 , \gamma _2 : (- \varepsilon , \varepsilon ) \to M$ two smooth curves with $\gamma _1 (0) = \gamma _2 (0) $ and

\[ \gamma _i ' (t) = V (\gamma _i (t) ) \]

for all $i \in \{ 1, 2 \}$ and all $t \in (- \varepsilon , \varepsilon )$. Then $\gamma _1 = \gamma _2$.

Solution:

Let

\[ \varepsilon _0 : = \sup \{ t \geq 0 \,\vert \, \gamma _1 (s) = \gamma _2 (s) \text { for all } s \in [0,t] \} . \]

We need to show $\varepsilon _ 0 = \varepsilon $. By continuity, one has $p : = \gamma _1 (\varepsilon _0 ) = \gamma _2 (\varepsilon _0)$. Pick a chart $(U, \varphi )$ around $p$ and let $\hat {V} \in \mathfrak {X}(\varphi (U))$ be as in the proof of Theorem thm:e-and-u-manifolds.

For $j \in \{ 1, 2 \} $, define $\alpha _j : (- \delta , \delta ) \to \mathbb {R}^n $ as

\[ \alpha _j (s) : = \varphi ( \gamma _j (\varepsilon _0 + s) ). \]

Then $\alpha _1 (0) = \alpha _2 (0)$ and

\begin{align*} \alpha _j ' (s) & = \varphi _{\ast } ( \gamma _j ' (\varepsilon _0 + s) ) \\ & = \varphi _{\ast } V ( \gamma _j ( \varepsilon _0 + s ) ) \\ & = \hat {V} ( \varphi ( \gamma _j (\varepsilon _0 + s) ) \\ & = \hat {V} ( \alpha _j (s) ). \end{align*}

Hence by Theorem thm:flow-line-uniqueness, $\alpha _1 = \alpha _2 $ in their domains of definition. This means that $\gamma _1 = \gamma _2$ slightly beyond $\varepsilon _0$, contradicting its maximality.

Combining Theorem thm:e-and-u-manifolds and Theorem thm:flow-line-uniqueness-manifolds, we conclude that for any $p \in M$, there is a maximal open interval $I_p \subset \mathbb {R}$ and a unique flow line $\gamma _p : I_p \to M$ with $\gamma _p (0) = p$.

Escape of flow lines If $ T : = \sup I_p < \infty $ and $t_i \in I_p$ with $t_i \to T$, then the sequence $\gamma _p (t_i) $ diverges to infinity. This means that for any compact set $K \subset M$, $\gamma _p (t_i )\notin K$ for $i$ large enough.

Solution:

Assuming the contrary, there is a compact set $K$ containing infinitely many of the $\gamma _p (t_i)$’s. By compactness, after taking a subsequence, we can assume $\gamma _p (t_i) \to q$ for some $q \in K$. Let $W \subset M$ containing $q$, $\varepsilon > 0 $, and $ \Phi : W \times (- \varepsilon , \varepsilon ) \to M$ given by Theorem thm:e-and-u-manifolds.

Pick $i$ large enough so that $ T - t_i < \varepsilon $ and $\gamma _p (t_i) \in W$. Define

\[ \alpha : (t_i - \varepsilon , t_i + \varepsilon ) \to M \]

\[ \alpha (t) : = \Phi ( \gamma _p (t_i) , t - t_i ) . \]

Since both $\gamma _p$ and $\alpha $ are flow lines of $V$ and they agree at time $t_i$, they agree in the intersection of their domains. This allows one to define $\gamma _p$ up to time $t_i + \varepsilon > T $, which is a contradiction.

Flows on compact manifolds Let $M$ be a compact manifold and $V \in \mathfrak {X}(M)$. Then there is a smooth map $\Phi : M \times \mathbb {R} \to M$ with

For all $x \in M$, one has
\[ \Phi (x,0) = x . \]
For all $x \in M $ and $t \in \mathbb {R}$, one has
\[ \frac {\partial }{\partial t} \Phi (x,t) = V(\Phi (x,t)) . \]

Solution:: If the flow is not defined for all time, then a sequence of points of $M$ diverges to infinity. This is imposible if $M$ is compact.

If $\Phi : W \times (- \varepsilon , \varepsilon ) \to M$ is the flow of $V$ and $f \in C^{\infty }(M)$, then

\[ Vf (p) = V (p) f = \frac {d}{dt} \Big \vert _{t = 0} f ( \Phi _t (p) ) . \]

Let $f: M \to N$ be smooth, $X \in \mathfrak {X}(M)$, and $V \in \mathfrak {X}(N)$ such that $X$ and $V$ are $f$-related. If $\gamma : I \to M$ is a flow line of $X$, then $f\circ \gamma $ is a flow line of $V$.

Solution:: For each $t \in I$, one has
\begin{align*} (f \circ \gamma )' (t) & = f_{\ast } ( \gamma ' (t) ) \\ & = f_{\ast } X(\gamma (t)) \\ & = V (f (\gamma (t ))) . \end{align*}